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Applications of Picard-Lefschetz theory: Landau singularities, ramification of the volume function, sharpness of shock fronts

Lecture



Applications of Picard-Lefschetz theory: Landau singularities, ramification of the volume function, sharpness of shock fronts, algebraicity of surface potentials, the number of independent generalized hypergeometric functions.

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Lectures and tutorial on "Theory of singularities and catastrophes"

Terms: Theory of singularities and catastrophes

Introduction: the essence and applications of singularity and catastrophe theory. Thom's seven elementary catastrophes
1.1. Hadamard's lemma and its corollaries
1.2. Critical points of functions, the Morse lemma. Classification of singular points of smooth functions and maps
1.3. Stability of nondegenerate critical points
The division theorem and its corollaries. 2.1. Multiplicity of a function of one variable
2.2. The Weierstrass division theorem as a fundamental result of singularity theory: its essence and purpose
2.3. Newton-Puiseux series
3.1. Germs in singularity theory
3.1. Germs in singularity theory
3.2. An algebraic digression (algebras, rings, ideals)
3.3. Jets and flat functions, versal deformations, stable singularities and sufficient jets. Jet calculus and transversality theorems. Stratified sets
4. Critical points of functions: 4.1. Corank and codimension of a degeneracy
4.2. Critical points of corank 1 and 2
5.1. Formal power series
5.2. Smooth functions with special properties
5.3. Three lemmas on smooth functions for the study of singularities of «continuous» objects
6. Singularities of plane curves
6.1. Normal forms of germs of curves
Examples and construction of plane curves
7. Multiplicity of germs of functions and maps. Multiplicity of functions Rn → Rn and Rn → R
7.3. The Malgrange preparation theorem
8. Left-right equivalence, contact equivalence
8. The philosophy of general position
9. Singularities of maps R2 → R2: the fold and the cusp
9.2. Singularities of codimension 3
10. Singularities of maps Rn → Rn
11. Singularities of maps R2 → R3
12. Proof of the division theorem
13. Implicit differential equations
14. Envelopes of families of solutions of implicit differential equations. Catalan's paradox
The topology of the nonsingular fiber near a singular point of a complex analytic function. Milnor's theorem. Vanishing cycles, Dynkin diagrams
Bifurcation diagrams: swallowtails and caustics. Generalized braid groups. Local monodromy of complete-intersection singularities. Picard-Lefschetz formulas
Ramification of integrals and the Gauss-Manin connection. The global monodromy group
Applications of Picard-Lefschetz theory: Landau singularities, ramification of the volume function, sharpness of shock fronts
Thom polynomials, non-removable singularities, and characteristic classes. The Maslov index and its generalizations. The universal complex of singularities and Kazarian's theory
The topology of discriminants and of the space of nonsingular maps. Smale-Hirsch-Gromov type theorems. Invariants of knots and of nonsingular plane curves
The local algebra of a singularity. Reduction of function singularities to normal form. The Milnor number, modality. The Newton diagram. Filtrations in the space of germs. Quasi-homogeneous functions and maps